$f'$ is a function, so it has a derivative of its own: the second derivative $f''(x)$, read aloud f double prime of x, is the derivative of $f'(x)$. Differentiate once more and you have the third derivative $f'''(x)$, f triple prime. In Leibniz's notation the second derivative is $\dfrac{d^2y}{dx^2}$, read aloud dee two y by dee x squared, where the 2's count how many times you differentiated: they are not powers of anything. For $f(x) = x^4 - 2x^2$: $f'(x) = 4x^3 - 4x$, $f''(x) = 12x^2 - 4$, $f'''(x) = 24x$, the fourth derivative is 24, and from there every derivative is 0. Each round lowers every power by one, so a polynomial of degree $n$ is wiped out after $n + 1$ rounds.

The trap in the name: $f''$ means differentiate twice, never multiply $f'$ by itself. $(f')^2$ is a different function with a different graph; for $f(x) = x^3$, $f''(x) = 6x$ while $(f'(x))^2 = 9x^4$. What $f''$ measures is how the slope itself is changing: positive where the curve is steepening upward, negative where it is bending down, and the next chapter reads whole graphs from that sign. At a point, $f''(a)$ is a number, found by differentiating twice and substituting last: for $f(x) = x^3$, $f''(2) = 12$.